Lommel function explained

The Lommel differential equation, named after Eugen von Lommel, is an inhomogeneous form of the Bessel differential equation:

z ^{2} (d ^{2y})/(dz ^{2}) + z (dy)/(dz) + (z ^{2} - \nu ^{2)y} = z ^{\mu+1} .

Solutions are given by the Lommel functions s_μ,ν(z) and S_μ,ν(z), introduced by,

s _{\mu,\nu} (z) = (\pi)/(2) \left[ Y _{\nu} (z) \int ^{z} _{0} x ^{\mu} J _{\nu} (x) dx - J _{\nu} (z) \int ^{z} _{0} x ^{\mu} Y _{\nu} (x) dx \right], S _{\mu,\nu} (z) = s _{\mu,\nu} (z) + 2 ^{\mu-1} \Gamma\left( (\mu + \nu +1)/(2) \right) \Gamma\left( (\mu - \nu +1)/(2) \right) \left(\sin \left[(\mu - \nu) (\pi)/(2) \right] J _{\nu(z)} - \cos \left[(\mu - \nu) (\pi)/(2) \right] Y _{\nu(z)\right),}

where J_ν(z) is a Bessel function of the first kind and Y_ν(z) a Bessel function of the second kind.

The s function can also be written as^[1]

s _{\mu,} (z) = (z ^{\mu})/((\mu - \nu +1 )(\mu + \nu +1 )) {} _{1F} _{2(1;} (\mu)/(2) - (\nu)/(2) + (3)/(2) , (\mu)/(2) + (\nu)/(2) + (3)/(; - 2) (z ^{2})/(4) ), where _pF_q is a generalized hypergeometric function.

See also

• Anger function
• Lommel polynomial
• Struve function
• Weber function

External links

• Weisstein, Eric W. "Lommel Differential Equation." From MathWorld—A Wolfram Web Resource.
• Weisstein, Eric W. "Lommel Function." From MathWorld—A Wolfram Web Resource.

Notes and References

1. Watson's "Treatise on the Theory of Bessel functions" (1966), Section 10.7, Equation (10)